Degree sequences forcing Hamilton cycles in directed graphs

We prove the following approximate version of Pósa’s theorem for directed graphs: every directed graph on n vertices whose inand outdegree sequences satisfy di ≥ i+o(n) and d+i ≥ i+o(n) for all i ≤ n/2 has a Hamilton cycle. In fact, we prove that such digraphs are pancyclic (i.e. contain cycles of lengths 2, . . . , n). We also prove an approximate version of Chvátal’s theorem for digraphs. This asymptotically confirms conjectures of Nash-Williams from 1968 and 1975.